Final position of an object after aplying several horizontal forces I'm given an object with a starting position. Several horizontal forces are applied to this object one after the other for different amounts of time. How can I determine the final position $x$?
 A: If you know the total force as a function of time, then you know the acceleration as a function of time also
$$ a(t) = \frac{\sum F(t)}{m} $$
Now you find the velocity and acceleration using direct integration
$$ v(t) = v_0 + \int a(t)\,{\rm d} t  \\
x(t) = x_0 + \int v(t)\,{\rm d} t  
$$
If the forces are constant then you can convert the integral into sums using $\int F \,{\rm d} t = F_1  t_1 + F_2 t_2 + \ldots $ such that for example after the first force
$$ \begin{aligned}
a_1 & = \frac{F_1}{m} \\
v_1 & = v_0 + a_1 t_1 \\
x_1 & = x_0 + v_0 t_1 + \frac{1}{2} a_1 t_1^2 
\end{aligned} $$ 
with $x_0$ and $v_0$ the starting position and velocity. The for the next force
$$ \begin{aligned}
a_2 & = \frac{F_2}{m} \\
v_2 & = v_1 + a_2 t_2 \\
x_2 & = x_1 + v_1 t_2 + \frac{1}{2} a_2 t_2^2 
\end{aligned} $$
and so on ...
A: You should consider Newton's 2nd Law: $\vec{F} = m\vec{a} $ and relate the forces acting on the object to accelerations. Once you know the accelerations for different periods of time you have broken down the problem into a simple kinematics one of an object moving with constant acceleration.
